Whakaoti i te {l}{y=-x+7}{y=6x}

$$\left. \begin{array} { l } { y = - x + 7 } \\ { y = 6 x } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=7$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$y+x=7,y-6x=0$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$y+x=7$$

Me tango x mai i ngā taha e rua o te whārite.

$$y=-x+7$$

Whakakapia te -x+7 mō te y ki tērā atu whārite, y-6x=0.

$$-x+7-6x=0$$

Tāpiri -x ki te -6x.

$$-7x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$-7x=-7$$

Whakawehea ngā taha e rua ki te -7.

$$x=1$$

Whakaurua te 1 mō x ki y=-x+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-1+7$$

Tāpiri 7 ki te -1.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=7$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y+x=7,y-6x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1&-6\end{matrix}\right))\left(\begin{matrix}1&1\\1&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-6\end{matrix}\right))\left(\begin{matrix}7\\0\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-6\end{matrix}\right))\left(\begin{matrix}7\\0\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-6\end{matrix}\right))\left(\begin{matrix}7\\0\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-1}&-\frac{1}{-6-1}\\-\frac{1}{-6-1}&\frac{1}{-6-1}\end{matrix}\right)\left(\begin{matrix}7\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{7}&\frac{1}{7}\\\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}7\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{7}\times 7\\\frac{1}{7}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=6,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te x ki ngā taha e rua.

$$y+x=7$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$y-6x=0$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$y+x=7,y-6x=0$$

Me tango y-6x=0 mai i y+x=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+x+6x=7$$

Tāpiri y ki te -y. Ka whakakore atu ngā kupu y me -y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$x+6x=7$$

Tāpiri x ki te 6x.

$$7x=7$$

Whakawehea ngā taha e rua ki te 7.

$$x=1$$

Whakaurua te 1 mō x ki y-6x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$y=6$$

Kua oti te pūnaha te whakatau.

$$y=6,x=1$$